Combining relations via relational composition Let R be a relation from A to B, and S be a relation from B to C Let a  A, b  B, and c  C Let (a,b)  R, and (b,c)  S Then the composite of R and S consists of the ordered pairs (a,c) We denote the relation by S ◦ R Note that S comes first when writing the composition!

Combining relations via relational composition Let M be the relation “is mother of” Let F be the relation “is father of” What is M ◦ F? If (a,b)  F, then a is the father of b If (b,c)  M, then b is the mother of c Thus, M ◦ F denotes the relation “maternal grandfather” What is F ◦ M? If (a,b)  M, then a is the mother of b If (b,c)  F, then b is the father of c Thus, F ◦ M denotes the relation “paternal grandmother” What is M ◦ M? Thus, M ◦ M denotes the relation “maternal grandmother” Note that M and F are not transitive relations!!!

Combining relations via relational composition Given relation R R ◦ R can be denoted by R2 R2 ◦ R = (R ◦ R) ◦ R = R3 Example: M3 is your mother’s mother’s mother

Representing Relations Epp section 10.2 CS 202

Representing relations using directed graphs A directed graph consists of: A set V of vertices (or nodes) A set E of edges (or arcs) If (a, b) is in the relation, then there is an arrow from a to b Will generally use relations on a single set Consider our relation R = { (a,b) | a divides b } Old way: 1 2 3 4 1 2 3 4

Reflexivity Consider a reflexive relation: ≤ One which every element is related to itself Let A = { 1, 2, 3, 4, 5 } 1 2 5 3 4 If every node has a loop, a relation is reflexive

Irreflexivity Consider a reflexive relation: < One which every element is not related to itself Let A = { 1, 2, 3, 4, 5 } 1 2 5 3 4 If every node does not have a loop, a relation is irreflexive

Note that this relation is neither reflexive nor irreflexive! Symmetry Consider an symmetric relation R One which if a is related to b then b is related to a for all (a,b) Let A = { 1, 2, 3, 4, 5 } If, for every edge, there is an edge in the other direction, then the relation is symmetric Loops are allowed, and do not need edges in the “other” direction 1 2 5 3 4 Called anti- parallel pairs Note that this relation is neither reflexive nor irreflexive!

Asymmetry Consider an asymmetric relation: < One which if a is related to b then b is not related to a for all (a,b) Let A = { 1, 2, 3, 4, 5 } A digraph is asymmetric if: If, for every edge, there is not an edge in the other direction, then the relation is asymmetric Loops are not allowed in an asymmetric digraph (recall it must be irreflexive) 1 2 5 3 4

Antisymmetry Consider an antisymmetric relation: ≤ One which if a is related to b then b is not related to a unless a=b for all (a,b) Let A = { 1, 2, 3, 4, 5 } If, for every edge, there is not an edge in the other direction, then the relation is antisymmetric Loops are allowed in the digraph 1 2 5 3 4

Transitivity Consider an transitive relation: ≤ One which if a is related to b and b is related to c then a is related to c for all (a,b), (b,c) and (a,c) Let A = { 1, 2, 3, 4, 5 } A digraph is transitive if, for there is a edge from a to c when there is a edge from a to b and from b to c 1 2 5 3 4

Applications of digraphs: MapQuest Not reflexive Is irreflexive Not symmetric Not asymmetric Not antisymmetric Not transitive Start End Not reflexive Is irreflexive Is symmetric Not asymmetric Not antisymmetric Not transitive

Sample questions Y 23 24 25 26 27 28 Reflexive Irreflexive Symmetric Which of the graphs are reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive 23 24 25 26 27 28 Reflexive Y Irreflexive Symmetric Asymmetric Anti-symmetric Transitive

Matrix review We will only be dealing with zero-one matrices Each element in the matrix is either a 0 or a 1 These matrices will be used for Boolean operations 1 is true, 0 is false

Matrix transposition Given a matrix M, the transposition of M, denoted Mt, is the matrix obtained by switching the columns and rows of M In a “square” matrix, the main diagonal stays unchanged

Matrix join A join of two matrices performs a Boolean OR on each relative entry of the matrices Matrices must be the same size Denoted by the or symbol: 

Matrix meet A meet of two matrices performs a Boolean AND on each relative entry of the matrices Matrices must be the same size Denoted by the or symbol: 

Matrix Boolean product A Boolean product of two matrices is similar to matrix multiplication Instead of the sum of the products, it’s the conjunction (and) of the disjunctions (ors) Denoted by the or symbol:  

Relations using matrices List the elements of sets A and B in a particular order Order doesn’t matter, but we’ll generally use ascending order Create a matrix

Relations using matrices Consider the relation of who is enrolled in which class Let A = { Alice, Bob, Claire, Dan } Let B = { CS101, CS201, CS202 } R = { (a,b) | person a is enrolled in course b } CS101 CS201 CS202 Alice X Bob Claire Dan

Relations using matrices What is it good for? It is how computers view relations A 2-dimensional array Very easy to view relationship properties We will generally consider relations on a single set In other words, the domain and co-domain are the same set And the matrix is square

Reflexivity Consider a reflexive relation: ≤ One which every element is related to itself Let A = { 1, 2, 3, 4, 5 } If the center (main) diagonal is all 1’s, a relation is reflexive

Irreflexivity Consider a reflexive relation: < One which every element is not related to itself Let A = { 1, 2, 3, 4, 5 } If the center (main) diagonal is all 0’s, a relation is irreflexive

Symmetry Consider an symmetric relation R One which if a is related to b then b is related to a for all (a,b) Let A = { 1, 2, 3, 4, 5 } If, for every value, it is the equal to the value in its transposed position, then the relation is symmetric

Asymmetry Consider an asymmetric relation: < One which if a is related to b then b is not related to a for all (a,b) Let A = { 1, 2, 3, 4, 5 } If, for every value and the value in its transposed position, if they are not both 1, then the relation is asymmetric An asymmetric relation must also be irreflexive Thus, the main diagonal must be all 0’s

Antisymmetry Consider an antisymmetric relation: ≤ One which if a is related to b then b is not related to a unless a=b for all (a,b) Let A = { 1, 2, 3, 4, 5 } If, for every value and the value in its transposed position, if they are not both 1, then the relation is antisymmetric The center diagonal can have both 1’s and 0’s

Transitivity Consider an transitive relation: ≤ One which if a is related to b and b is related to c then a is related to c for all (a,b), (b,c) and (a,c) Let A = { 1, 2, 3, 4, 5 } If, for every spot (a,b) and (b,c) that each have a 1, there is a 1 at (a,c), then the relation is transitive Matrices don’t show this property easily

Combining relations: via Boolean operators Let: Join: Meet:

Combining relations: via relation composition Let: But why is this the case? d e f g h i a b c d e f g h i a b c

How many symmetric relations are there on a set with n elements? Consider the matrix representing symmetric relation R on a set with n elements: The center diagonal can have any values Once the “upper” triangle is determined, the “lower” triangle must be the transposed version of the “upper” one How many ways are there to fill in the center diagonal and the upper triangle? There are n2 elements in the matrix There are n elements in the center diagonal Thus, there are 2n ways to fill in 0’s and 1’s in the diagonal Thus, there are (n2-n)/2 elements in each triangle Thus, there are ways to fill in 0’s and 1’s in the triangle Answer: there are possible symmetric relations on a set with n elements